In this paper, we study some supercongruences involving the sequence \( t_n(x)=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}x\\ k\end{array}}\right) \left( {\begin{array}{c}x+k\\ k\end{array}}\right) 2^k \) and solve some open problems. For any odd prime p and p-adic integer x, we determine \(\sum _{n=0}^{p-1}t_n(x)^2\) and \(\sum _{n=0}^{p-1}(n+1)t_n(x)^2\) modulo \(p^2\) ; for example, we establish that \(\begin{aligned} \sum _{n=0}^{p-1}t_n(x)^2\equiv {\left\{ \begin{array}{ll} \left( \dfrac{-1}{p}\right) \pmod {p^2},& \text {if }2x\equiv -1\pmod {p},\\ (-1)^{\langle x\rangle _p}\dfrac{p+2(x-\langle x\rangle _p)}{2x+1}\pmod {p^2},& \text {otherwise,} \end{array}\right. } \end{aligned}\) where \(\langle x\rangle _p\) denotes the least nonnegative residue of x modulo p. This confirms a conjecture of Z.-W. Sun.
